4,558 research outputs found
Emergence of scaling in random networks
Systems as diverse as genetic networks or the world wide web are best
described as networks with complex topology. A common property of many large
networks is that the vertex connectivities follow a scale-free power-law
distribution. This feature is found to be a consequence of the two generic
mechanisms that networks expand continuously by the addition of new vertices,
and new vertices attach preferentially to already well connected sites. A model
based on these two ingredients reproduces the observed stationary scale-free
distributions, indicating that the development of large networks is governed by
robust self-organizing phenomena that go beyond the particulars of the
individual systems.Comment: 11 pages, 2 figure
Quantum quenches and thermalization on scale-free graphs
We show that after a quantum quench of the parameter controlling the number
of particles in a Fermi-Hubbard model on scale free graphs, the distribution of
energy modes follows a power law dependent on the quenched parameter and the
connectivity of the graph. This paper contributes to the literature of quantum
quenches on lattices, in which, for many integrable lattice models the
distribution of modes after a quench thermalizes to a Generalized Gibbs
Ensemble; this paper provides another example of distribution which can arise
after relaxation. We argue that the main role is played by the symmetry of the
underlying lattice which, in the case we study, is scale free, and to the
distortion in the density of modes.Comment: 10 pages; 5 figures; accepted for publication in JTA
Preferential attachment in the growth of social networks: the case of Wikipedia
We present an analysis of the statistical properties and growth of the free
on-line encyclopedia Wikipedia. By describing topics by vertices and hyperlinks
between them as edges, we can represent this encyclopedia as a directed graph.
The topological properties of this graph are in close analogy with that of the
World Wide Web, despite the very different growth mechanism. In particular we
measure a scale--invariant distribution of the in-- and out-- degree and we are
able to reproduce these features by means of a simple statistical model. As a
major consequence, Wikipedia growth can be described by local rules such as the
preferential attachment mechanism, though users can act globally on the
network.Comment: 4 pages, 4 figures, revte
Preferential attachment in growing spatial networks
We obtain the degree distribution for a class of growing network models on
flat and curved spaces. These models evolve by preferential attachment weighted
by a function of the distance between nodes. The degree distribution of these
models is similar to the one of the fitness model of Bianconi and Barabasi,
with a fitness distribution dependent on the metric and the density of nodes.
We show that curvature singularities in these spaces can give rise to
asymptotic Bose-Einstein condensation, but transient condensation can be
observed also in smooth hyperbolic spaces with strong curvature. We provide
numerical results for spaces of constant curvature (sphere, flat and hyperbolic
space) and we discuss the conditions for the breakdown of this approach and the
critical points of the transition to distance-dominated attachment. Finally we
discuss the distribution of link lengths.Comment: 9 pages, 12 figures, revtex, final versio
Mean-field theory for scale-free random networks
Random networks with complex topology are common in Nature, describing
systems as diverse as the world wide web or social and business networks.
Recently, it has been demonstrated that most large networks for which
topological information is available display scale-free features. Here we study
the scaling properties of the recently introduced scale-free model, that can
account for the observed power-law distribution of the connectivities. We
develop a mean-field method to predict the growth dynamics of the individual
vertices, and use this to calculate analytically the connectivity distribution
and the scaling exponents. The mean-field method can be used to address the
properties of two variants of the scale-free model, that do not display
power-law scaling.Comment: 19 pages, 6 figure
Spatial preferential attachment networks: Power laws and clustering coefficients
We define a class of growing networks in which new nodes are given a spatial
position and are connected to existing nodes with a probability mechanism
favoring short distances and high degrees. The competition of preferential
attachment and spatial clustering gives this model a range of interesting
properties. Empirical degree distributions converge to a limit law, which can
be a power law with any exponent . The average clustering coefficient
of the networks converges to a positive limit. Finally, a phase transition
occurs in the global clustering coefficients and empirical distribution of edge
lengths when the power-law exponent crosses the critical value . Our
main tool in the proof of these results is a general weak law of large numbers
in the spirit of Penrose and Yukich.Comment: Published in at http://dx.doi.org/10.1214/14-AAP1006 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Modeling the Internet's Large-Scale Topology
Network generators that capture the Internet's large-scale topology are
crucial for the development of efficient routing protocols and modeling
Internet traffic. Our ability to design realistic generators is limited by the
incomplete understanding of the fundamental driving forces that affect the
Internet's evolution. By combining the most extensive data on the time
evolution, topology and physical layout of the Internet, we identify the
universal mechanisms that shape the Internet's router and autonomous system
level topology. We find that the physical layout of nodes form a fractal set,
determined by population density patterns around the globe. The placement of
links is driven by competition between preferential attachment and linear
distance dependence, a marked departure from the currently employed exponential
laws. The universal parameters that we extract significantly restrict the class
of potentially correct Internet models, and indicate that the networks created
by all available topology generators are significantly different from the
Internet
Socioeconomic Networks with Long-Range Interactions
We study a modified version of a model previously proposed by Jackson and
Wolinsky to account for communicating information and allocating goods in
socioeconomic networks. In the model, the utility function of each node is
given by a weighted sum of contributions from all accessible nodes. The
weights, parameterized by the variable , decrease with distance. We
introduce a growth mechanism where new nodes attach to the existing network
preferentially by utility. By increasing , the network structure
evolves from a power-law to an exponential degree distribution, passing through
a regime characterised by shorter average path length, lower degree
assortativity and higher central point dominance. In the second part of the
paper we compare different network structures in terms of the average utility
received by each node. We show that power-law networks provide higher average
utility than Poisson random networks. This provides a possible justification
for the ubiquitousness of scale-free networks in the real world.Comment: 11 pages, 8 figures, minor correction
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